Thanks! Consider my concern withdrawn. I wasn't quite certain exactly to what degree small effects might factor into the design. At some point I'd love to end up doing some experimentation on this all myself, but I am currently involved in several other projects so mostly I'm just a lurker. Keep up the good work!

- That was known from beginning (and always was a difficulty for purely thermal mechanistic) : for some modes there is "no significant net thrust" without dielectric while same mode with dielectric exhibits thrust. This is known for TE012. The absence of thrust without dielectric was for "some very early evaluations", the experimental plot is not published. A TE012 mode with dielectric and thrust is reported in Brady's report.....

What is new is the reporting of Poynting vector plots, to show the Poynting vector field for different modes and their significance.

Although Dr. White published several papers predicting that the force on the EM Drive would depend on the Poynting vector, for example,

Quote from: p.10 of Brady et.al.'s "Anomalous Thrust Production from an RF Test Device Measured on a Low-Thrust Torsion Pendulum"

Consideration of the dynamic fields in the ¼ wave resonance tube shows that there is always a net Poynting vector meaning that the RF launcher tube assembly with dielectric cylinder common to both the slotted and smooth test articles is potentially a Q-thruster where the pillbox is simply a matching network.

I have not seen any published calculations by Dr. White and his group of the Poynting vector field for the EM Drive. There are no Poynting vector plots shown in the paper "Anomalous Thrust Production from an RF Test Device Measured on a Low-Thrust Torsion Pendulum" by Brady et.al. and there have been no Poynting vector plots shown by Paul March in our threads either.

Consideration of the boundary conditions for the Poynting vector seem to also have not been mentioned in the published literature of the EM Drive.

I have not seen the following previously mentioned in any EM Drive report:

Quote

Since the components of the electric field E parallel to a copper surface (either the wall or the bases) must be zero at the surface, the Poynting vector component perpendicular to a copper surface (either the wall or the bases) must be zero at the copper surface (either the wall or the bases) .

Let me repeat that: the Poynting vector component perpendicular to the small and the big bases of the truncated cone must be zero at those surfaces (must be zero at the small base and must be zero at the big base).

For a Transverse Magnetic (TM) mode the Poynting vector parallel to the surface doesn't have to be zero. Actually, as the images show, in some cases the maximum Poynting vector occurs at the wall for a TM mode, and for a TM mode the Poynting vector at the wall must be parallel to the wall.

On the other hand, for Transverse Electric (TE) modes both components of the Poynting vector (parallel to the wall and perpendicular to the wall) must be zero at the copper surfaces (either the wall or the bases) . For TE modes the Poynting vector is zero at all copper surfaces: zero at the walls and zero at both of the truncated cone bases.

The following points do not appear to have been previously made either:

Quote from: Rodal

Examination of the Poynting vector radial component shows that for this particular mode (TE012) without a dielectric, the Poynting's vector is self-cancelling and hence it is not a surprise that NASA measured no thrust force for this TE012 mode without a dielectric, since according to NASA Eagleworks' own theory (relying on Poynting's vector as per Dr. White's papers) there should not be a thrust force without a dielectric for mode TE012 because Poynting's vector self-cancels for this mode.

....However, for other modes (TM311 for example), Poynting's vector is not self-cancelling, but it is pointed towards the small base. This justifies the fact that Shawyer communicates that he is presently not using a dielectric, since a dielectric does not appear necessary for certain modes.

Using finite element numerical method to numerical analyse the classical Maxwell equation of electric field of the idealised conical resonator, to obtain the model and practical of the distribution of the electric field of the cavity under 1000W. By analyse the properties under different modes and the different properties. Calculation show that under the four modes, TE011, TE012, TE111 and TM011, the quality factor of TE012 is highest and with highest thrust, followed by TE011. With the Small End of the cavity unchanged, the quality factor and thrust decrease with the increase in the Large End

Prof. Juan Yang writes that her Finite Element calculations show mode TE012 as having the highest thrust (without dielectric). My exact solution calculations show that the Poynting vector fields are self cancelling for TE012 without dielectric.

Furthermore NASA Eagleworks experiments confirm the self-cancellation of the Poynting vector field for mode TE012: NASA, using the truncated cone without dielectric, and excited at the TE012 frequency, "measured NO significant net thrust": the complete opposite of Prof. Juan Yang's conclusion, but in full accordance with my calculations of the Poynting vector distribution for mode TE012.

....BTW what would be the heating profile for TE012, with and without dielectric ?.....

THERMAL PROFILES FOR DIFFERENT ELECTROMAGNETIC MODE SHAPES FOR A TRUNCATED CONE EM-DRIVE

Please find attached below the square of the norm of the magnetic field at the big base of the truncated cone without a polymer dielectric section for the following electromagnetic mode shapes:

Cyl. TE012: this is the mode shape we were discussing. This is the mode shape discussed by NASA Eagleworks in the "Anomalous..." report that was tested with and without a polymer dielectric section. When tested without a polymer dielectric section they measured no thrust force.

Cyl. TM212: the only mode for which the temperature profile was reported by Paul March, as measured by an external thermal infrared camera (with a polymer dielectric section inside the cavity). This is also a mode shape that takes place, without a dielectric near the frequency that @Mulletron is planning to test.

Cyl. TM311: This is also a mode shape that takes place, without a dielectric near the frequency that @Mulletron is planning to test.

The temperature distribution through the circular cross-section should match the (square of the norm of the) magnetic field distribution because the surface gets heated by the magnetic field induction heating producing eddy currents in the copper.

The frequency obtained by the Finite Element method ranges from 1% (for TE012) to 2% (for TM311) less than the exact solution. The Finite Element solution is more stiff (the FEA converges from below). Higher modes (like TM311) require a finer mesh to obtain the same level of accuracy as lower modes.NOTE 1: The experimental results obtained by Paul March at NASA Eagleworks (see attachment to http://forum.nasaspaceflight.com/index.php?topic=36313.msg1327406#msg1327406) with a thermal infrared camera pointed at the big base of the truncated cone for mode TM212, fully confirm that heating of the big base of the truncated cone is due to the magnetic field heating the big base by induction heating: producing eddy currents on the base. See these images

Note 2: I have not yet had the time to write the computer program to calculate the exact solution for electromagnetic modes in the truncated cavity with a polymer dielectric insert, but based on the fact that both the reported thermal IR camera measurements by Paul March and the COMSOL Finite Element calculations by NASA were for TM212 with a polymer dielectric insert in the EM Drive cavity, and that both the measured temperature distribution and the Finite Element calculated distribution are in perfect agreement with my exact solution calculations for the EM Drive cavity without the polymer dielectric insert, it looks like the heat distribution for mode shape TE012 should be the same with and without the polymer dielectric section.This conclusion is also supported by my exact solutions for the cylindrical cavity with and without the dielectric.

This conclusion is also supported on theoretical grounds, since the polymer dielectric insert is homogenous and isotropic HDPE (or at least transversely-isotropic), and hence it should not affect the mode shape circular cross-section distribution.

... it seems like you're saying is that we should discard the Poynting vector? ...

I have not see any other calculations and plots of the Poynting vector for the EM Drive.

I showed results that for mode shape TM311, with the cavity without the polymer dielectric section, the Poynting vector points clearly towards the small base half of the time and towards the big base the other half of the time. This mode has not been reported as having been experimentally tested yet.

I showed results that for mode shape TM212, the mode shape presently being tested by NASA Eagleworks in a partial vacuum, with the cavity without the polymer dielectric section, the Poynting vector practically self-cancels. I predict that NASA Eagleworks would measure practically no thrust with the polymer dielectric removed. NASA Eagleworks reported that when the Nylon 6 bolts melted, they lost thrust force. This appears to be consistent with these results.

I showed the fact that boundary conditions for the transverse electric (TE) electromagnetic mode shapes all Poynting vector components vanish at the walls of the EM Drive and the fact that for mode shape TE012 without the dielectric the Poynting vector is self-cancelling. This is consistent with NASA Eagleworks results and supports NASA Eagleworks preference for the transverse magnetic (TM) modes over the transverse electric (TE) modes.

When I have the time I also would like to calculate and show the distribution of the divergence of Poynting's vector as well as the vorticity of Poynting's vector for the truncated cone's EM Drive.

.....In your opinion, do you see any correlation between magnetic field strength at the large/small end as seen in the Comsol plots and the measured thrust levels? .....

One should not seek a correlation between the magnetic field vector by itself and the thrust force, even in MagnetoHydroDynamics, or for the EM Drive, because the magnetic field vector oscillates like a harmonic function, hence it spends as much time going in the negative direction as the amount of time that it spends going in the positive direction. Ditto for the electric field vector by itself.

Instead, one should consider the quadratic functions that appear in the 3+1 spacetime form of the (contravariant) energy-stress tensor Tμν, which contains the Poynting vector (S) and Maxwell's stress tensor (σ) . Quadratic functions such as E^2 and B^2 that vary as (Cos[ωt])^2 or (Sin[ωt])^2 have a positive time average. However the quadratic function (Cos[ωt])(Sin[ωt]) is still self-cancelling over a period since it spends as much time with negative magnitude as with positive magnitude. Poynting's vector varies as (Cos[ωt])(Sin[ωt]) = (Sin[2ωt])/2 and therefore oscillates at twice the frequency of the electromagnetic fields and it self-cancels over a period of oscillation

Both the energy and momentum conservation laws can be simply expressed in relativity in terms of the divergence (in 3+1 spacetime) of the energy-stress tensor Tμν

The energy-stress tensor Tμν is also neat because it directly shows how energy and momentum are inexorably tied together in relativity: lack of conservation of momentum is tantamount to a lack of conservation of energy, as @frobnicat and others have repeatedly pointed out.

Could the discrepancy between your calculations and the Chinese results be due to a typo or translation error? Given that both happen, and keeping in mind the now resolved dispute between your numbers and those of Eagleworks.

Could the discrepancy between your calculations and the Chinese results be due to a typo or translation error? Given that both happen, and keeping in mind the now resolved dispute between your numbers and those of Eagleworks.

I don't think that this particular, specific pointed discrepancy (that mode TE012 without a dielectric resulted in no significant force in the NASA experiment and that it also has a self-cancelling Poynting vector field according to my calculations, while Prof. Juan Yang wrote in her 2010 paper that TE012 was the mode shape giving the highest thrust force without a dielectric) is due to a translation error. I also doubt that it is a typo because that statement is made in the body of the article as well as in the conclusions and it can also be ascertained from the plots shown in her 2010 paper.

I don't think that the specific pointed discrepancy (that mode TE012 without a dielectric resulted in no significant force in the NASA experiment and that it also has a self-cancelling Poynting vector field, while Prof. Juan Yang wrote in her 2010 paper that TE012 was the mode shape giving the highest thrust force) is due to a translation error. I also doubt that it is a typo because that statement is made in the body of the article as well as in the conclusions and it can also be ascertained from the plots shown in the paper.

Wild speculation, then:

The Chinese, if I recollect correctly, are pumping a lot more energy into their device than Eagleworks.

I have seen repeated mention here before this results in a lot more heat, and the frequency changes with heat. So maybe this morphs TE012 into something else?

I don't think that the specific pointed discrepancy (that mode TE012 without a dielectric resulted in no significant force in the NASA experiment and that it also has a self-cancelling Poynting vector field, while Prof. Juan Yang wrote in her 2010 paper that TE012 was the mode shape giving the highest thrust force) is due to a translation error. I also doubt that it is a typo because that statement is made in the body of the article as well as in the conclusions and it can also be ascertained from the plots shown in the paper.

Wild speculation, then:

The Chinese, if I recollect correctly, are pumping a lot more energy into their device than Eagleworks.

I have seen repeated mention here before this results in a lot more heat, and the frequency changes with heat. So maybe this morphs TE012 into something else?

That's the first thing that jumps to mind, if one has to speculate. Also she uses different equations to calculate the force. Her equations also lead to another contrarian statement she makes in her 2010 paper (and in the conclusions of that paper) that keeping the small base at constant diameter and keeping the same axial length, increasing the diameter of the big base decreases the thrust. (I wonder whether she still believes those calculations as her later experiments involve truncated cones with increased -rather than decreased- cone angle of the EM Drive).

I don't think that the specific pointed discrepancy (that mode TE012 without a dielectric resulted in no significant force in the NASA experiment and that it also has a self-cancelling Poynting vector field, while Prof. Juan Yang wrote in her 2010 paper that TE012 was the mode shape giving the highest thrust force) is due to a translation error. I also doubt that it is a typo because that statement is made in the body of the article as well as in the conclusions and it can also be ascertained from the plots shown in the paper.

Wild speculation, then:

The Chinese, if I recollect correctly, are pumping a lot more energy into their device than Eagleworks.

I have seen repeated mention here before this results in a lot more heat, and the frequency changes with heat. So maybe this morphs TE012 into something else?

That's the first thing that jumps to mind, if one has to speculate. Also she uses different equations to calculate the force. Her equations also lead to another contrarian statement she makes in her 2010 paper (and in the conclusions of that paper) that keeping the small base at constant diameter and keeping the same axial length, increasing the diameter of the big base decreases the thrust. (I wonder whether she still believes those calculations as her later experiments involve truncated cones with increased -rather than decreased- cone angle of the EM Drive).

But there are alternate explanations as well. For example, many of her statements in her 2010 paper are based on her calculations, and not based on experiments. It could be that her calculations (in that specific paper) that TE012 should give the highest thrust are incorrect. It could also be that her calculation that she was testing TE012 mode was incorrect: there are no IR thermal images of her EM Drive to verify what mode shape she actually tested, the geometrical dimensions are not given (hence not possible to double check independently) and there is just not enough experimental detail given (unlike the NASA Eagleworks tests were a lot of information was given) to speculate and much less to know for certain.

Based on what you're saying, I'm going to make another (hobbled) sense antenna, for now it will be just a solder cup. Then I will repeat the same VSWR test to see how things change.

Here's before and after shots showing the difference between matching drive and sense antennas compared with just a solder cup as the sense antenna.

I was able to confirm that when using just a solder cup as the sense antenna, it makes no difference in VSWR whether or not the sense antenna is terminated, left open, or shorted.

Differences include finding less resonant frequencies (because the sense antenna is also a resonant structure), and resonant frequencies change.

I noted that the output as measured at the sense antenna was attenuated by 32db compared to the input, which is good. That means more energy is staying in the cavity, instead of being coupled out.

So essentially, when drive and sense antennas are the same, the cavity is a really efficient cavity filter. Data from the previous run showed performance as good <2db loss from input to output. Since both antennas are the same, this ensures that energy is coupled into and out of the cavity with ease......not what we want here.

I also sacrificed a connector by nibbling down the solder cup 1mm at a time to see if the VSWR got better or worse. That testing showed no improvement whatsoever, actually it made things worse.

A bare solder cup is optimal.

So my most likely candidate for unloaded testing is 2413.5mhz. I have no clue what mode is being excited here.

I'm not sure MJ Pinheiro does consider this project related to the concept I was talking about. Here is a quote from, "The Physical Basis of Electromagnetic Propulsion", http://arxiv.org/abs/1502.06288, "However, DEF don't work on the basis of EM radiation, like is supposed to propel the NASA and Chinese projects. IT is supposed to work on the near-fields propelling effect;". Abbreviations are, "deferential electromagnetic force (DEF)". If there is DEF going on maybe it would answer the question how propulsion could be happening inside the cavity without the emission of radiation. I'm trying to find a link that indicates an overall poynting vector imbalance possibly connecting the two ideas but so far I'm having trouble putting my finger on it. There are some other related papers I was looking over such as "Dimensional Analysis of Thrusting by Electromagnetic Inertia, http://arxiv.org/abs/1502.01917". It looks like I should read up on this also, "http://ntrs.nasa.gov/search.jsp?R=20110023492"

Clearly this frequency ( 2.4135 GHz) falls right in the range of the calculated frequency for mode Cyl. TM311, if the dimensions of your truncated cone are within 1% of the assumed dimensions (Big Diameter=11 inches, Small Diameter=6.25 inches and Axial Length=9 inches).

It is the best mode to excite (in that frequency range) with a cavity lacking a polymer dielectric, because this mode shape (TM311) has a clear Poynting vector. The other modes have zero Poynting vectors in the longitudinal direction of the truncated cone.

As to why the measured frequency (if correct) is closer to the Finite Element Analysis (FEA) solution, here are possible reasons:

1) Internal dimensions of truncated cone may be 1% larger than the assumed internal dimensions in the exact solution analysis (Big Diameter=11 inches, Small Diameter=6.25 inches and Axial Length=9 inches).

2) Flatness of the big base and the small base. The exact solution assumes spherical section surfaces for the bases while the Finite Element solution assumes them to be perfectly flat. This can be shown to make a small difference, and its sign (increasing or decreasing the frequency) depends on the electromagnetic mode shape.

Do Hall thrusters use Xenon or can they use a more common fuel since Xenon is rare. Massive tonnage to Mars or the Moon over time will deplete a lot of Xenon. I was thinking something like hydrogen, oxygen, or some other more common element for mass use in space.

You have the math for the dimensions of the cavities as a function of the resonant frequencies. Have you attempted to cast the dimensions in terms of the wavelength of the drive frequency, in closed form? That would be interesting to me in formulating models in Meep since I currently am modelling the geometry using wavelength as my base unit of length. Of course, maybe I am only asking for a simple division which I can easily do for each individual mode.

In discussing the Poynting vector you rely on the ideal boundary conditions being zero at the cavity walls. To what extent do the shear forces resulting from copper being "not ideal" modify these assumptions? 0%, .001%, 1%, 10%? And how large might those left over shear forces be?

Do Hall thrusters use Xenon or can they use a more common fuel since Xenon is rare. Massive tonnage to Mars or the Moon over time will deplete a lot of Xenon. I was thinking something like hydrogen, oxygen, or some other more common element for mass use in space.

See the Electric Thruster thread. There are other options. the ELF thruster in particular can use anything. But that is where you will find them The answer though is yes. Ion thrusters can use stuff other than xenon.

...In discussing the Poynting vector you rely on the ideal boundary conditions being zero at the cavity walls. To what extent do the shear forces resulting from copper being "not ideal" modify these assumptions? 0%, .001%, 1%, 10%? And how large might those left over shear forces be?

There are many boundary conditions. I presume you are referring here to the electric permittivity of copper, effectively assuming copper's permittivity to be Infinite so that the Electric Field E tangent to the copper surface is zero (to satisfy continuity of that vector component of E ). It looks to me the effect of impurities should be extremely small, much smaller than 0.001%, see:

Continuity of the vector component of E tangent to the surface must be satisfied (see http://en.wikipedia.org/wiki/Interface_conditions_for_electromagnetic_fields ), so that if one were to pick a finite value for the permittivity of copper, it is straightforward to quantify the effect of assuming it to be Infinite, and one then readily arrives at the conclusion that the effect is extremely small (closer to 0 than to 0.001%).

OK, so that means that by modelling the cavity as perfect metal, I'm not missing by very much. That is, perfect metal is a good assumption for copper at the frequencies under consideration. I wonder how much higher frequency I can use before that assumption starts to break down. I know that copper becomes transparent in the mid THz range, but I think it maintains its character up to low THz frequencies. (1 or 2 THz).

I'm looking at the option of modelling the cavity at much higher frequencies, relying on the linearity of Maxwell's equations to relate to 2 GHz. The reason is that Meep runs much faster at higher frequencies where the dimensions measured in wavelength are smaller. Its kind of like trading spacial resolution for time resolution but they are way out of balance at 2 GHz for these cavities. What is the opinion on that idea?